On Polynomials which Have All Their Zeros Outside the Circle |Z| = R in the Complex Plane Citation Yood, Bertram (1939) On Polynomials which Have All Their Zeros Outside the Circle |Z| = R in the Complex Plane. Master's thesis, California Institute of Technology. doi:10.7907/8j5j-zs42. https://resolver.caltech.edu/CaltechTHESIS:12182024-164018576 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1939 Record Number: CaltechTHESIS:12182024-164018576 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12182024-164018576 DOI: 10.7907/8j5j-zs42 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16938 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 18 Dec 2024 19:26 Last Modified: 18 Dec 2024 19:44 Thesis Files PDF - Final Version See Usage Policy. 9MB Repository Staff Only: item control page

whose ~t·n-·o,a nre t h&"l complex nu1Ytb ere" In th1$ p t:aJ>Olt' we s h.nll be inte:,r-estad c hiefly :b:1 pOlj-rM>mi11le whic1?1 htlV8 all theAs, P/":J ~ ... . 4· e < be ~ p0l-;y:oond.el of· deg~ee _,,, • ,, Z: •.::"" :Pbe.n e. nec0esery and. aui'f'ie:1en t cond.1 tion for all t Le zel?O.B - of Pt-al to ~oe O~J ts ide $01'.r:.e c1relt; b0 1-0. Pr>Oof': fJeeeesi,~:p ''loe!'Oll I• fr~, . is that ! f all t he ze ros of Ptclo.r:e out side 1~I:::.R, R>o, t hmi _,z @t'O ie not .a of I ¥:/-=z fl(,, ~ >'di:> 'l'hlilt'e S.l."e of rti!:)~ L' • ..12. 1n ~ll ?-t seroe of. !'It/ ,. "- -,:o. 011. (.l ot tbo:m, s &1 o(. must be,. .in absolu te v ~lue, closes t 01" am close €t:S m:ry of' t t e o ther Wzeros 0 to t h.<.~ quta.r1.tity zero. '-Ne ce.n ehooee ~ es ,:my iiiwtber· E.Hii tire.fyi.rig t h.0 in.0que.,li t7 "< R. <: /of./. This eott,pli:1 tes t he pxioot of t he t heoran:.11 e. necesmar1f and suff1e1ent oondit1-ons tor all .,_ . the ~eroa ot Pt~,= ,--=o ~I>,·~ c. to have all 1 t s zeNe outtd:t'ie t he 11;.;e now stl')e-k theor-em io ee f oll owe -~ Let / (-c-) =- i; a , i: <' ,,~.::, • rtJ:1en f Ir::./ ha.a t\ l l 1 ts rioQt& 5r ~ttb:l~ ti'1e 'U!lit 01riel$ 1t il.nd ~nly it t!~ (?1- 1J" - d$GNO i, /r) == ~lln(li~.!ol ~ { 4":. J(~ ~oz~ J(-f) j J - htta ~11 itlil St~·!!!()~ w-1 tl:iirt th4!'i 1.1.nt.t.. el.rel~. a,~d 1t ttl$0 ~ /ti,,,..J>Jqo/. \'JG i't-t;'>W 1~t,~1~ t@ /"1~) = z: 1,,- z We 1:.' • ~'":::.o .r1i❖s t: fftili:e t-l'i.G Oh tm~i1t O,f "'itJUr•t.r~l)l • .... ~ be· R'. ~,: Wb:)$e 1-·=o • of rt~J. ~i~ new OOt}$:1\d~j) t h0 i1ol:;fr~mb:il >. I 1?,? J,· K'. "ji "f'l't/ = ~ ,.::~ ~H.'i~ S,QN)I o.t "'' 'I ) $,f+f] t l':~ . ,,._ ll'tHl;i~1-~~H;lQl,g h~ _. · I ,. -';, ' of t 1r.llU,i) of { ';I ,) • t.t· " rt 'a J be&.$ $'ll l 1 te ~.e s:"O • t.if J t;,h! ti t he 'U,'! l t e! )\"•(fh) , t t;~l!, (pl1 J b tu» tlll !te ~o~e out~id$ t t.~ mit etl"'el~., ~n d Pf~ t&ao all 1t s • .,. P{-e.J:. / 1= I =- k a:we 1° /hJI Jb EI,'° "l:. i to ID.tave all 1 t-t,1 teros but$1'3l.~ t bt, et).~(tlO ,·:o ::> 1, {r.} :: $!!.e~~ ct J "- /{'"/ t { /,~ 3-t Pf-l:) 1ft" c) - "'· @Utf$Sd1t:, Ir/-::.. R aeeot!p11.et:ie4 f;r1 et mo5t fr'- ste,1;~ .. !("f ( i ) !'o:ft ftknf R J • Thia Oa!, bf.t ~JP en0 ~~a t.~s:t 3-,1-2:) by j, l~J r,11111. iM:1t"V·e n l l ! ts ie:~1 Wli \;bl.ft thfl S ~!':tl:t:ft $ i::}1e,rJtt'f;.~}~ a~ii~ w-:1! . t 011,e10 tt ~ c ~li'ti~} f n /11-,..} de~~ea J<)l'f-r:ltf)tid~,l t-"tl~o t1ite t t-tii ~ttJ~o• t l:i&:fe, arHl !!1 ~l®o a sat1efi@'t1. :!nf>qt~¢\l it1 ~t ! t;e ~..oe:f.tl ttl~n t• la t'Hil:r·t ~.m 'fti~ .i ne~bl~mti· prov14et1 t L!e la 8(!.!.t!l1t'iG<i, oe:n tootl 1 0 1~·tOltli~l, ~-,d -~~ f¢,lr'th. 1n n f:!.n,s.,~ llWffri,.~!" of• ~tepl r~~)'tffi Qn~ ein:i t~'.il whtJt!.~ t• o~ not t\ll ·t,,;:1(, 3tnFfltiJ 0t Pf~J e.~ -O. ()nt•l(lW' flie ·S ittmt,!lo:n. in Wh!eb t.h(iJONiw 2 l~~'tltf!J& ~ ~t1Sft\~to:r,, l:tut n0t pNr.itt.tl@~lJ.J " '8.$ to the@ etUl$ of u. us 1.~ t,J')t,)t-3 ~l~Ot>eti.Oti, 117 b$C$U.1lH'! tt, g1?$f~ no el'U9 tttt\ rae,tt dflttf)t~ oune1vee ~ t h ! .~ qu~st1ot&. St'?@ftl .¢ itlte~1ir1t1t!,t)1)$ ~t !l n:va to tz~ fO'Ul:'lti ffi t b~ llt@'l"tl• tu~e tiJrtou..~ i~ tl.1l.f)lti$ O!" tb.o .,..foll cwt.rig ? (l) 2 tli ~ •fi~'l ~eN:s ()If" q, eooplfJw.: nU11ibe1?& .sit·e to be· 'J ,~d tom"'i d in t h~ K~ltil\}S. Mt@~~!nf!>-t'i bJ th® e1\r'OJ.e$ w1 th eent;.t)r fl,t ' --J... orlgb1 ~tl i'tadi:t. /It r,., - ., . , \'llh,e" fh1 = a. ~/ ~ { / :: / ~ / a;;• / -> • .. ~ / '-'-~ f ) l'1 ~ H~ ( J";: :: ,.,. _L .. , I ::~ /., . --., I'::- I ) J) ... z;'°Q,· :ft l,tl'i} ·~ t~!t~ f f~J -=- t~i ~ll the ~~~- flf ,~, ,~, p { r J ~-.,a in tb.0 nt~tiulttll ~-~ ~ 1.·. -'" -t ~ Qo S ot.,, t . !{¢.\~'l i k!$ba1,,g ef"t TJti~ :r. df;e t J:ig~ itl G.lU.ntt dti LO$t.~ ~oo OO.$t' .Alg~br'6:~~ohan t,l.~l!.elllftli (1006) See t; ,. B. Kell$h itu• J't1U:f"tlt'tl ~A ft'iith Dee '1 tli€~1" . 1~u1~u•if:o Rtir~d.. Vol. ~e i~ ." 34~• ~3'5 0 l.!rii1tei; iee. a(:!J~e d.t'oo. folvno• Vol, '1 f . 1t fli•l71 (1916} . ..,. . ' . . . . .·.. . . , ~ Th is gives us another d e termination of R, . ,.. . -~ .. namely MY J. s ">- quan tity s maller than Other dete rmina tion~ will be gi.ven below; ' • ' With regard to the various determinations or R whieh ean be given, the following theorem is of interest because it gives ., . ... .. . , . , an upper limit to the v a lue of R. . I ~ ThE3oreni 3. Let ft~ J = &1;11 l,i c' with /,u :! 0 - Then R < I::. J-;;:. .Jr« tJu_ Jiu.-.,.. / ..t, / ••• I .I,.. I = I ·~ I Ft11 J.J-1,~-·',l,J,.,. whence either/ 0 } "'' ~1-- p { 2:-). .J. I= } "'~' ~ • •• --/.IM/=- IJ::.f;r ,1.. • $\Pat least one of / a',,• I < //,"~I:!: .. This completes t h e proof of t he theorem. Theorem 3 can be strengthened b7 the finding of a necessary and sufficient condition for all the zeros of circle J ~, = / Plrl to be on the t; /¾._ For t his purpose we use a theorem of I. Schur (1) to the s. . effect that t h e . zeros of an )"I~ deg ree polynomial f (~/~ ~tlL· i:-' l .::d are all distinct an.d on the unit circle if and only if r a" (v-==-1, . . . , ... .>,,,..) fl. -Pt-" .2 has all its zeros within t h e uni t circle.t", To apply this t heorem to the problem at hand we consider the 1- given f (-1-1 ==- 2 ( l- c i ' =" If f/e-J has no nult1ple zeros, than. we apply the following: Make the change of va.r1,..able Tben f (-c J:: 6) I '-J J _::. f b., Ji R -- I b,.. . z-=- R 'a- , /,~- /? ; /) ,· 1 and ?tz.J he.a all its zeros on t he circle /=t-1=-R if and only it /j (:, J has all 1 ts z e ros on the unit eircle. The zero of {f( ~J are all dis tinet s:ince t hose of are distinct. Pf~) (1) I. Schur Journal Fur Ma th. Vol. 148 P. 136 (1918) P(-i:-) 0fl!it~ r.J~ t @:~ t .~ to:w itul t 1~1n &tHOfh ~~1t1:ti 6(-11 .o f f'fi:-J ~n d p//2-J ('t.>11 ~ ~11d1 e l;,l~r-ithi't'.\ .. e~7 ). t l~ p,ol,nomlsla 7,;~J > Ille O,.Ch~D• '~ · ~ e~n~i.d~l~ 6/e/ 'ft ?t~J it$~1t h ~$ no t',tul t ~pl$ h "•s , t.~$t¢1tM:l Of. ,~,rtin,@$ /'{~/ f<.1flt" t~/i!'V tni nlJ. 1t~$ llJf.11~!J Oft f{:; t.ei:¢3 t tv'$ t le J to&" t httt p:f"()p-i!1t"tl1'+ ~'l~Hl rl,;rn t,1~, th~ c.c .». ot ~I~ I;t 6t~J i'W:e t'ef!t"l 1Il pl~l ~tlP.ie, ~.ni 6- 'I~J • or. f l2 J tn ~1J~ tl'tifltt11.a.tor.i :abit,ve ~~011 <>f t lJ$ pt;>!~o1ni~le tt 6,(2:-J !t 6, 1-l J 'k~€;;$ n ~ ~,ul ti!.f,ll e 8~~ O◊t1 $$.d@r hdH~~~tl }.!$& ·~n1.1lti~l1~~ fli~~$ fO:Nfcl Dit~/ 'th a o.e.»•. <,>f 6;(2-J imd Ir,' ( ~J antii 't~$t ttie ~11-tio:rntals 111 Qtt&e ~,~, b~.& ~.(.) ll1~1 t !.p1e •ei~s. It 6J:,,\ l:M~tl wnltipl@ $ :( ~""1$ '11a p:rti>o~ed 1n t tae, \'ile'.9 heN bl• 1~11.UJ we dJ.eat{j(t• (H)I'J.~i,fl:tifil' O'Ui'' $@0 t t.~t t t ~" 1$ t~o 1t'ls~ ~"'l gm'lt,rall. ty tt we ji tv.~n Pf•J to !':ii-\\V:& i'lO nml.tlpl i$ IHlrQ$ .• t hltt ,fati.ve~ ue- t~h t} t 1$0N<t"t : d!.,s tJinct t11Jei~$:). Then ,:.:1, nt)O$$:~f,i r•1· aru-1: .i n,1;f f:'le!en t eo:1.::dtt!Qt1$ t.f 'iiQt 19 {~/ :bi~•~ £iJ. l lte $tlJO-S 011 t,h.e e.1~,010 1 ~I-== I(= 1" J,,. t.,.,_" ,). 0 {; 'r7)~ = /.,'>o I:,., J!:.J ¾ atr1e ( v c. '~ . . . ,, n-t J a..tl d,, ~~ /1«I~ '•"" ~ t;> 1~1~ ,fh;,ll '}[ tr:i eonr,&etl~ .t with (.t,Jndt tl:)1n. .2 1 of th1$ ·t be~~otei ,,. n~{~eru'l ear1;.v- m14 snJ.ff:l .t1.:l~:nt eorJd.iti('f.~1~ ro~ .2.' to be ~aitt~ti~Mi nr~ g ,.v~ri ~ l,)1<11$ ). ~.bove• (Jtbe~ :neee$aa:irv ~n(! $ttf f'te '1$.i~ t t1tJnd! t-!one C(Yt.lld ~lJ.110 be us~. {l) ( l} I i1chll~1 .r~um~il tt~tj,~ Nt~M'1on~; atteb . '-t) , . 148 • p., 1 34 O. tilB) g :l:~(!Jt:{ ~,(rta~ tl1(i$tt) tlie .i;r1.»f~ ,tit ~tn:,rtJif't ~ t-;l ,.f!ti~e UZ?'~ &tk:tU" li1;!t'ei i;,Ot'{:t·ttHlfh, tin e $.;ef~l l,t,,~J~.t Ol H3 d~J;~. to (J!I•OJ~~ ,(1) • :t,tt ~ q/ I ~ 7 , • • .,1 H 1 >- >/!> ' 7 . , . > f<>A.>o w.1d ,J =- }f~ ( .I - tf.1 / - y, ... ,, K- >i , Jl ) /Jt~{,l-~ 1 (4-r, _µ. !:;i~t a,,. a~ J . . . 1 ···.1H-1t~A) J a,. 1 a,. .,ti.o be .tt:t'b1tr-&,~ "~.l 01:" *~~pi~~ 1\Ullb ~ti• t@~ wh! ti.b ~~Cb. a, f O ~~, Fl- ~ H~ { / a~/ , J ~/ J ~ Jt~ ( I:~ I I J ~l(}n ~&.Oh r,•0 (11-t /. l,(..,t X -' + 1 ,. • J /;; f J :: / ) I J, ... ~ I 4 ~ I 12: I } 4 (A'; I t ot t h$ ~~~:i . eti ()n q,. , " I- • f (f._,. K '4 -1- ct o : O Jts-t. < /1:J < (l) { ..L J.- l ,, S ,,, I i·Ji n. C:n~~1ft : VtHfal li~i1.,1m0t,Th:Itil,c't¼1.g (ie~ flt1t~~~-ui v ot-a ~~eJO. ~ r.:t0l~ ~~ t h. ,rour1r:a1,,. V~il .• 4 0 l "Htt®fJ '19~5 • .(195'1) ! bf:: .~ 1lt}1t1!n,1;t; 1f) e me.tH'Jild f~i• ilk'lt~~m:hd.:t'.1~l t>p~e1~l e ,i1>~0, tni t 1$ t •~tt:ie~ intero~rtf,t)h • ft ~ . a Vt1!7' (l) tt /C•L >/C,/>··· > Jc.,.,/> o W'I¢l !l = }1-{ c,,_ J I - C"/ I c;,_, I - ICv/ Th$?'.i ~tl l th~ t:'JMlt crt tr-se ( tJ.= I, J. ~ ••• ' /ft J } p@lyo~td,tii1 /(11~ C"+C,1<-+···+-C.,,,..'/',., 4N @utat.d.$ t ~if1, cii"ele I~/.:::- • ~ Aw.t~tJ1(;,:r d i0t0r.tt!'! tJ(.\tton t>f' ti tl$ er.1rrb.t~it"HHi ili B ~~11J~l? b7 \l'l: !.@it ~on t~1,fl.$ t l~(t t"O$'ftl t t h~t t he ab"SOli'Ate "' f t l \l0:C o:r tii8 l0Nti Qf ~le ,ol','!~.W:ital a, r ( BN «i¼.l i ~OfMft it'a:$4 ,,' ~\nS11el,u.t~& (I} z; &".:o 1n the int~•~ tiol ( x,, )I,_) ,,b$N r, f~~tt y1 &~ t ;,::,~ ~.1n.f,!l~ f.+tuilt1'1e ~ira-..0$ ot t .bt) po!t!ffl$~1a:t, /ll . . . l x~ + Jq,.._,/ x."-1 ~ ... -+ /Q, Ix - /a,J-. I a,.J r,. --Jttk"4fx~-,_ ... -/a,{x - /<lo/ nep,~c t: lV$1 y ., 1.~i e quel')®tl\')tl tJf t hi~. upr,<1~ ~::>m1:tl of ta~~$ of' -n pol~io.~!,e;l 11~ ~~i t.tiv!Ostia'!l~,tetl $1u.el i itt~f!','.';l oa" f \1:llJ ~ 1d eitt&n ~ri-v~ly tJu.111 t be ,qu.:4!lu t.tc,n tut" 1{).w-et" bt}m,t"i • whi ,et. w0 l'~i.'la c.onii:J,d-et'$d. I t le ' d!,e ~)'t:tll~t\.(l}l1or.t von d~~n it'Ol7;t1€)1'¥.1 €t ~nd tt4r (l.) ~ $f) K , t: h"ltlt~tH ~ Potiin0~e Ulo.n.. ftillek\l ~0,tt1. J'tlt~fth Vol. 55. ( Ult ~ ) P. i~}\•2~ (ta) tm~~1ii)l utnt,& 1l'h• Su~ un~ i1.t~t.t it1(liet.n 41 tm. 1i1@()~ID-0 d.0 Uuwr1! .t tt. St:ill.- ~~ot . t!iel,. 1<\ et}d ff;(jc,i~ •. Vel l ~l. P ~ llJ)•.:)_£1 (19-~) These upper bounds ave to be found 1n mony and va:r. .u, It we reetr1et ()urselve.a to· polynomi als pl&ee.s 1n the l1te1•ature. with veal coet:f'ic1en ts the nu.rr1ber of upptr bounde tound in t h$ 11te~atu:re inc:reaeea en.orI11ou&ly. Sueb. upper bounda aN CW♦n tcimd 1n elementary texts on.<t he t ll$0t'J' or e qua t .i ons. {l) However we ah$.ll consider here t he general ease where the coefficien ts ot the polynomial may be aompl$X n:um:bers. 1. /v1 +- , where "1 U) give11 b-y Koe11igeberger as e1ted above . (P., 5 ) f =- (l} 1-1 Q.1 /L I~ -t-•·· r , ...M ~0 I).. See Dieks<>n, First •ur&e in t he Tt.ieor, ot Eqaat1ona Pi(J.(5es 21, 22. . stve..n ·01 w,ell•ne:r i?i.s ef,t~d, a~0. ( i• . I)" •• ~ 1e eJtt~noo ~t Cli~~..,g tlt·r.t eitMi a'b~e-. ,. 11t:i& ~~,te~i~n ct .rt ~onelutt~ aa fi1t•d ~1~¢J, (1 } -J- t'it l_ ~ _, J • · • -#- fl"M "144 a; Z b l , O • ~,..d, ~ 11:I < t 1- /t:i.,- 1 I.., + I ~c 'i,r- + .. · -1- /4.,,.-Q..,...,/'L + ,a,,..J 'fh& iltlNti or z. ., ~ q, e,. -· .,.. ... . ,. . q ._ -11 110 on ~i" wlthb\ t bo 1 r , ,_ t. e1Nl• I'!:/~/«,/+ ft:t.,__J-/.+ · ·· +/fi,../; v.. c.,.. +0- 1 ~ " - ' ,- • .. ~)e &Wolu.~ v etlti• o f' ·t h$ ~ei1ioe ot i':>;@;,s t he ,1p-per l!,id ,t • { / 1- ( IQ, J .'1!20i• f ~ 0. II- p -,. (!) ..!. I ii~/ ,.,_, J -1 1 a-- 1- • • • I l+}I} r -,.a.,.,_ j L 1-1-1' (i) a. ~ ~ l ~niS.~1 in '1 .. b~.$ ~ $ t }:1$ ~~r litim.i t Gf t h~ abl~lut4 v~l. ue ~f l ts 1..-N1e . ~~ 111aJtb'~ (ljf t,'he niur68@rt ~.,. ~..,. ( :: J1c / le ~ ,, >-~ ... \tbere >.I(.> o 1 >. ,-, .,.. ""_,._ +... -1 >:;.' "'"'·<'•.~ •4t 't.'PI-Q>1't'l'<, 4'!1,lt:fMo:~ ~..{l;YJ ~ ,... 5-- ....,._m ~~ '/K J ~ I (4),; ~, tilse tbJl ilr"',;.""!!-"':,-'•'",.,,,., ·,1/,,-1'\,B , w,~ , ,..i. ~ .W,\i;~j,~ ~W.'1P" I ~'+~I , I0.1.lf,'-f.f., ' J ... J J11.,. .• 1(,,,,. .• ,f.,._-a,.) - I f~-, la ,d (1, f._ • • • 1,..,)- 1 lfflflt~ f,>•.1 P1.--;,,,11_, ... ) ~ - , >• 10. 'ft1@ numbt1'!1 [ I ,-. /ti, /~ I- . . . 1- / 4"" /.,,.. J t-1n WvPt~,r, l 1rn!. t ~ :t~c. It) ( .$ ) ~ , { / c,.., ~ L) !s ,~,c. (', • l- -:. P, J .,_ .Then .6/ ( '3 J ::: ? '" f, ,·, '° 1=• hu5 ali itti z~ros J~tslda Form ·" ~ ~ s :, ·"- i ?.t)nt~lf (2) 11,Jnt,el C ;_ ;:fi;;t (~;;.~ r~ ~sr:c ·-,:i ~ :~:i '!(>l'ioiL U. 11u 't!·1. Vo l . J --~'-1rr1~1J.. l , ·-, /. \! .) . "- •· 41 l'/4- 'Y/6 (l·-131 ) 2 . Jll-Jl6 ( l ?J6) St~}has as its zeros t he reciprocals of t .he zeros of ~/ ~; and all the zeros of S<h} are within the unit circle .. a We apply the methods 1. to 12. above to the polynomial SI J and thereby get a quantity, B 1 say, v,hich is an upper limit to the absolute values of the zeros of such quantity ¥le tak e the least .J'f',J. Now B, 'f I B, that the se methods give us. • ,.L. Suppose S, < I Then the zeros of ~I~.) are outside/2-/= Ji'., • and hence the zeros of P(.tc) are outside lz:. J-::.. ~ /3 1 ~ J we have a be tter determination of If we obta ined. 8 1 =I f ► in place of R R, namely~~ 812. . I f, . R. We c an now proc eed as before And if /3 1 ( obtained as was /3 I ) is l es ;;;; than unity, we obtain a value determination of Thus if the method we give here do e s not enable , us to improve on our valu.e f'o:r l' sing . 8 which is a better . than either I-', ,-,... P2. We proceed in this manner as far as pos .: ; ible, that i s , as long as the 13'> obtained are l ess than unity. We now turn our a ttent ion to other que stions, to properties of polynomials whi ch have all their zeros outside t he circle Our first result is the Theorem 5. Le t f'(-z:/ =~·~c, ;t' I,,- X i b e a polynomi a l of degreen wi th all its zer·os outside the circle / t/::; /?_ ~.,. -, jt D • Then the polynomial of the ..,. -, . • .tr (,. -} - 1 '::: t Jt ::> ,,, degree .s (~)~..zz.j = ~ ( ~ - i.,J J,- ~ £ ,·::. 0 also has all its zeros outside t h e circle Proof: R <I* . / c-/.:::. I? We divide our ,)r oof into t wo c a::...~s; ca se l, ~ ~ ( c ase 2 · -- -- - - - - , , - - - - - - - - - -- - - -..... *This is unnecessary as the proof for case11 is equally valid for case 2. ;:;; 1.nC f!c· t h,) ' Z O'rOS Pf eJ -s.m:d or Q 17) .;J l'e R~1 , • !>,.._,._, J' ((+t) 1•:; I ttll lio within tbt~ from >.-I to . ... -, • 0 L (1, _,•) Jc• 'J'' , ·=-tJ z ~ ie t ,,.. l'• ~ r'1 J ::: ,·.::,, e" 'I ' C. we apt,ly the re:sl:.;.t t of 0~1.se one rind ha v~ .th/i t s r 'I 1 :::. ~ r~-'") c~· 1 •• -=- , . .t rt,.,; J(J·,, j ' , nomial .ltiitH 'Oi or s{ ; ) . all th~ Ja~ro~ f) f rr ~) l >,~ f(w-J ar~ t.hml out~~1d$ Ji;J-= R . ..,,, " = Z: (~ -,· JJ, Rilfo - Er,. -,·1 ,,. «f" '-; , ·-= ~ Th~ proof of th~orom. 5 .1~ naw oo,~iµleta .. \\ o a>l¼n 1 ot eour~<:1 1 ;J.Jtt(;fn-d tbit rtuhtl t ,,. .., :Z: (>"t'J J,· 7 ' fa ~/J-J tu1~ t;.,ll (f I~ 1 =- if' Wf~ to th~ 9olyn.::::,mial I ,·=o p,,._,,. #- ~ I~.,;,,->-/ lt!J J ,(;.) l'.'CJ~ r,;;ut1iict:0 /?/-;~ ohtt;1 in the :r'tlfs ,ilt tht1t , in thi~ Oli:.$e , ,a-:a. E ('Jt - /J{'Jt -c'~ I,,- z. ' M It) = I- ~· =-e, ha..s its zo-ros ftll out!lide /~/=-R. l..,_~ ~ 0 J }t :,, 3 Purthe:~ 1.f wee h.-:.\r:-~ thitt /Y{t } ::: ~ / :>t-J..-t.){1t.-1- ,· ){'>t - ,·) t,. C: l 1·:--0 ha$ oJ.l 1t5 z~re:i out;..! i1de It/= I{ . L-e-t ht> I Pit/ .::. ~,nd lfilt CU$~11Jn. abOV'{~ ..:Z: /,,· ~,: ~-~ "~· :/- o bo ,~ polynoi;tial of doi:retiJ ( i=- ~., 1:> ••• ., ~) • Th(lt:n the- µoly-nomi.al , L(-c.)= L 0 {i-«-)('J-G )• [.=.o h~s ull iti .zervi o;_ttside /:t.J:=-~. •• /1,-c';Jb,z: ( R </ ~b~ / • Hut th.is .~h::n-· o mttst litl outit idfJ / 2 /:::. R. J~!;-/ fie ~ t'/"bJ;,, Q "Ii!!::.)> ... , & 't ..,IJ :::.r e • Lfl!t f'/~1~ ?:,J,-2:c· J (!)/~/::, ~-=d \<'.it.~ h,rrrve tl~t .). 2:- L. I' 'le) G'/2: I - ) Pf~) & ''/2:-) Ir r~ ~ - ") l'IW (:J //a;, /L-</c/<-R hfas no z-1.:rra$ in fand l.ll>. 1:~Wi!icy ;. r [ p '1 l!) 0 ";-~.) - >- I'1 a- 1 Q ~ J 1i) />- ~ / t I c /?. ,,,,i'~, =- /#'fl .1 6.,. e + ,,.._,J ! a,..._, J na5 n.c; Zi, r~s in the nrmul.1.1@ But () 1 t. ?Mn .1. ct [ ( -1 '""",. __ i ~,- r'-) ("' ! o. £, 1--•J''•~ I - ,._ f.l'f•Ct'"'""'i · r'~-i' '""* - -•• _ V .J.. ~ 1'.~), ,... ~ ~ -Ii# J{r)= fJ. c / . ___ , ) / V'" "4"'- t:L ?'> -r;5,.5,i /1"~- 70 JIA,..., b., + I i I <I<. ,,,.,.,d ~~,.~:,-,,,..1ti'''' ''' .... '-.$ l- ; t.._;,. .).. ., 0,<'<.Y .,l, Z::f J,,,.a,.,{t-,}b,.., ':: 0 l-llf"4,..<.,.. ~-"' h«--, t (1) i • • in th,J 11nn;,11us ~ !ilt' ti~ (.i;. t 2 ')f"" ! &.J] ,I? tr'.J(,.,.!a,..,.~tf-~)!a,,.,_,) ' .: ,J h~.H, no (; t-'(~J= .-. / l.f..., +r~11-7' Jr,, __ , - ...,. ., ,.__ u.:,c;. rtt~ ·,, ,., •• ·., t,fl\ ,..,;.~ 'iil •,;. 1 J.t:i.,,_,ib; Ji' th :, t ~-~A-4 ->""""-J,._, J,.; l..,,... 4~ ?' J,. 7 ,, ).. ~II-- , .. +(W\. a..,,, (""' )I - ,,..,) b,..,} r -..+ I ~l',:;"'}1.0 ~:·u M ·:.~. ·t; l~®'Ji~. t i .t: i "tr 1 t:.T ~.)~1i-· ;·!;J.1. 1 ~- \.iT l rY$ 1.·:.~.h C 1:r:f.~ 5 Jj..fJr".,.E ' q;lJJ1tt::d_·J J$ .i,J.,ub:t\L~U 1.: z:o Vol, Jt$ ? . ~,, ... -100. ;~;€,l thf;:)rl:'rn:i ~r . () trr (:1·C~ tt~.t, f~~r 1 'IIJ. J. r ,. ,JV t_,:.t.9JJ J ~ .,. . It fl~)~ t •.:: o b,· z' Z ttz"•~ 'I+ . ..ilidll-..h~i.illfj W1 111'9.tff.(Ji, l'i~e mll !.ta tle1'0S oute'l de r21-= ~ J . • 0/~F"· Zet.· ~,· ht48 All tts ~f.H'Ol\ wt thin- 12-lfH"J c. ~, A, ,:,, ~ sc~)• •· .n t} l\\ll'OS l n trui &nriu1.us n.. , I c,, fl • ',Je r.a:ote t he l"0'$ Ul t 1.EJ b idepen•d'-tnt ot ·tJ).e nunfber,,s "-- ►.., t:1 0 ••• / t ho tlttmo Nill-Ult by tiik:ln.g a...__ ... -:..4_.._~;:.--•=Q_,==-o = Ci,_ ~ ""' ..,. 4 ,_ _, 2 ,... _, ~ C ~ _, Lr.£/_, -2-- -I- 4'-- - · J 0::1 tl l)&n6$ '?te obtain :riii d 1:, '-Y { ~ ) I :J./= ~ • Let ~ be tliflJ ,.~ ~itlve 11\m~b~~, !fjl•" poG1t1V$ t.~• I! I;_ I? t eger, .,{ and (3 ,:s~l'f t r.o n\Dbe~ J$Ufftl t httt tl~ pOl,nt»~11ttl (,\.,,.. _ ,-,.J ~o t- 1" - fl + tf • 1'!'J$1i . 1»1- " [ l-t 2 l,'7'1 [p. + ,;1\f._-2.J] ~ t ,·--,. - o ~-,.., J 1,,•. , -1-(-1[ 2 , •.,.. '"A -"J 1,,. f :Z.' -+I bae no zeiioG 1n ·t }\itJ· t'½nnulua / !:- / < I 2- I ~ R. • Xt in ltb8 theoi-ian1 ot ObNClitkeff we talt.$ htrt,rt> that- t 1~ ~l,nomS,tJl .1. r[/"/2J t ~- ~ f'lc:J,.,., l1t\8 no SOffl~ wt tl) ~ t....e~ .( )f / ~,: A ~ / ~ , =- ~ . w• ~,.,,.-,J + ('Ml\-.,,._) P l r , r""' e-~e~p-t 2:-.::. t1 • IPs.eto,:>1til tltlt ~ tiw 1:i 0l.~o~tl'll• 4,dl q\l~Gto\, 0 ~ \'!Otll d v ,t11 lall, •• t, . - ... (A,w,.-J.h)i 0 .:::.o, " ~ ~ . , L r· f'/i!-J=. ,:u1 I,· -2' ha& t,1ll lts B0N$ outel4E> tJM'J ~ ho F cJ· m;ut 1J 1111.~,el~ It/-:::.. IL.. ~id we $pplf theoi-er!l 1. ~ma ~ ar-~ r~e1t,1va lrltee~.u1J• hene• - ( ;>to ""' -1- ~ J bo ~tld l 1~t~e ~ -=- 0 l'ri. .O!t~$V~,, ttonov~,,. ~ > "> ~ d i\ ~ -1- ~ ~ 0 :I o 1a not. & tMl\'rO er OUJ• i')Ol,nOOt1~1.. 2 ~pr/ ~J-/;.:,,,-.+~Jl'l::eJ-::::: ?-t _ 2 µ. ~- - It !', .2,· J,. ~'-{A~., h.) ,.::1 o+t - "" ) J.,,- ~ • r;' /4· r.: c-u z, L•::; c:, >- :? o ta $;tl e,w1thl'J po&ltive I::n>.!itll$~, "'" ... .. --- Mil1@Ver iiktil~ a~,1t~r,- po~lt1ve mtc,ger' t l1en ;,. ,,.,... lt an orb1tmf1t peel• ~Jl .\ tlv~ . . . . . 'if:te eum el 1 tl},ef.l~ NSUl tt> i n the ~ . tbe~m '• tAt /~/~/: '~d Z/.,,-rc. be & t)()l~1eni1fll Of M,SN>f .... , (lfdflt . ¥ ~ - lb:tz ~ll 1 ttal 8(¾f'OfJ o-ut-~id(,, t :tie e,l rel~ I ~I~ I? arb1t~~r1{ postt1'tJ'e n~1ti $l:'* ♦ I"' :wttl:t Lir>t,.A -;r O ' be C\l.1. 1r:!:t~n th$ pcl.ynOt:ii&l z k .1.i!-/'r/r;-)- l""'1>+~) /'(cl.::: • (.2c.·-/'4--h)~i c' {":;.; bile no !1:$\NU!J w:1 th.it'! tli• elt•¢le I~,= p. We iiotl~e t h~t t h6 th$>NW lo$Vf16 ~p@'ill 'tho p()Gi{£b1lltf' of '1,,. L/,.c·:J<--'1)JiZ: ... l:u iYjf iij ~erotl on t;!')',e eta:1}¢14 tall t h0 te~a ot ~t:1y e1 mt;;e, . 2 1.~/=-k. I /2-,'-_,1,t->..J"· 2- 'aN outeide l~o 0!1~le ~/::: t~ E-1 n~b~~. , wtiorw At the . G 1 ta an tarh!t.r tlrily •~ll pooittve ·, TheoNm 9 e~ ·n ttir l)& ~pplf.ed the- t"JOl~tlOJit! i.1 f.J1,tlnS Ut) th:0 t"@,$'.\ll t t h~t the pol~i!t!al L 4' /:l-c'-;.,-,,..J{-2 i-p ~ ~) b ~ c' 4 • <·=o wh•N P-r le ~~- a~t.t1"m1,r po~l ttv• nurnbflJ" i~e no eeV()~ within t he ~lt"el.e /'l:/=X- €-1 * . 'f.ho pol~omtal eann-ot. htffe any ~eros on I l:/-= ~ as shown m • the Appendix. A. s1m1l&r sta~1»ent OM b,e rttJ.de for the polynom:b,1:le ot theo:rem 10, to t.Jt!S 11~w po.l.Jt~(.)id.al ~~4 obtlJ!ift. lldJ ~it• can ~!'Ji!Y th•N~ t Mfflll't tt•t ~ I ,Jl -_,1,c ,_ - .,,. )( 2. ,· '.A-r r X Le.·-µ - >...) /,,- ~ i' ,-=-., the elNle / ~ /.: X- 6, -&.._ W'.t)f)ft I"(.... Se tffl at-bt.tt~l'1t Poat.ti:•• s-► l\tnii;J:elt' en-d le ~-a a"lt~11, •~l ~a1t!~-e tlU&Yb~f', Pwoe0efSl'.t4J 11:;,. t hJ..lJ ~.,f%J it ,~ $MY to ~l;\;OW tl:~e&t tl1te pct),ril~P.U fls{-2J~ j ~) be• 2 ,• :Z: IT (J.. ~• ,-=:" wba-1~ " "/'A.. ..,,;, A , ~ . . . f S =- l / :J-~' • • :) ' ~A,...1• .. p<t}&itl\tfl 1iuitrbtlt-e,. $)re, til?b1t1"'&l"J bav~ n.o u,~ '".6l ,;,1Jt't'.ti.1rl tt..~ el;\"tf1c (~I c K - ( ~, .,. • • • + 6 .s ~ e, .J ••• wheN I ~., Qff e ~! tlt'{\l,1"''$.l'J' .f l~tJ,l ,~s;tt1re i:tureii:i~~a. ,'r!!ii$ :p~.f tft t tt1'1l etfat~ir;ertt l$ -~ we e~in in& ~ • ta , ..1 _ 1 , ~ Z: be a ;.-011no1mt1.ti or 4fJ3N• Pr~1 :::: c-::o 1'.~ ~' ~d, t i'.I ttll it$ ~6-l"Ofl ou,ta1do t l\$ @1rclG /el-== Jt. (Y' } 4 - , , ; - 4 . , , T t ~1 . . 1 nut $fr"nte!P' fltt~1t.$1.>tt.:y};1t, ~ie Tl1• ~i1 lJ!~ ?At tlil\MlltQ . izl4. A4i' . · ,@,eGJ tndbie,tton ,on S . , ,,,,14/\,,•·· • Let be tJ $~t\U$n,e e flt ~~?:,t.tJ"$.~/ :r;;;ot,1t:S:v6! ni~:dbeN. t~.e V@1:,n,Qtli1 ~~ 1£tc...... .s ' ,fsf~J-- ~ [T(;J..,'-,4"'-") J,c. ~ c C -,> {Ie~~_,.,.J ..-,;::./ :tt~ve no ll$l~~ itd,·t h.m t .h~ i~-i ,:·ele /~/=-~ . P('»r • - ~ •$it ~ r~, et~l,<t / -c- /-:. N , f2, / ' /l. , tt w~$ & l Y,iii~ ~,'b ovfl; tb-i ,t /~ I= K - ( ~ . + • • • .,. ~.s J @tnfit.11 ~• 1 ttv~ iiiw.ll~t:e. {.., f- · • • -,. ~., ~1on no ffN)• of ~ut 1:- -;1-, lllid tl~ aero 'l- ~ ~, wltt'd.n . _ ~- I~. I =- •cf" R&ra-J h~s f.:te whe" G , . , *Je <Hjg)ee• ) O&J, i,u"e '11!.t!~tl the etr,~10 . . . . , an {t¾ft,.,t"1.~11f ~ s &, ., ... ., "~ eo sxr~l. tbat =h ,:;. If Rs r~, l.:!l$ ~ithli, ti}fi e1ftle / t 1-:::: /i!.. - >.. 11~1 jJ!t11h~ tl]~ e.1N1• I 2:1 ==- t- >t • ~$'.i$G v;~ b,u,¥• "' ~s t ~ J o&.nnot ha1te aGros on or w1th1n le 1~ I{ • Se• Ap;pen.d1.x for this tmd for a simpl1tlod proof of' tl:iaorem 10. f;»t.t.t.~ uii.ctf.ffl'l, ~ltd t Ji~ t i'~~<1m 1& t,tf'.OVO@ . f~~\Jl bi dt:ft«'J:1"«1:,1t .r"11i. a, ~ut · td,lld,ltii.r t~. t tt60t~tr$ , . 9, 1.0 . l~f¥1 be obtfil.!Jt;~d, ff'Ci'~ ~not}),$)" t h $).f f~1 et ObN~t:i;'lu'>tr. (1 ) Tl~. ~llJONt t ~tf;,f;;$~ t -tWit 1.r t h fl 1l$·~e ot Pr~ • dog1•• ""' Et.N} Ot.at'8,$,t\e tu~ e!~elf> /2:1--:: fl. ~ ant! ·t h~,t 1f t hOtHl) of cM wt tl'dn tbe e:lrcle / ~, f ,... , ,a.. ' R Ii {i!,J t!.$ tlt'0$ ~4 1t • -<., -<, a.re a~,t~a~ t (non • 1~.u ll) ~ i~l(;)l'; n~tH'/$ $UOh tbiil S;nd ). l$: ~ OG®Pl $:& tiuir.te ~ euoh tltl$t · {. I - I-' I "> ( I- I al, I J 7 o ,.,.,~, t her1 t.he p()l.,nOEt~l /1~1~,Pf~i:-J &(-;!J-,. >R :,-) (xoT,~J ~ : ~ no $ !]jJ'$t.'J .,... lAt . o1, in t:t~ aruiu1ua ,,... ~ I -a-1 ,_ ~ ~• f'fcl ~ ~ Jl ~ c• a) /~J=: t'le· z; ce'":,i) ' 11}1; \:/$}St,• • --E.. ,, _ t J.JOC/Nm, cited ~~ve 0(-...,Jr~~ ' - - ttl€¾0 b l\$ all ltB ~e~• wttld;tt t '.h~ e!~·e le I~, ~ /j.. I /, h1 ,.,,,_, , "" I ~ ) :: {J r,--.J/~) {j, ~ {,,.,,·/,) • ..._ ., I A .,.AH. ""- ..z. ~ ~ (,w, -,I)! i--• I- J. I 6-. I (/,--,ro2.,r.J:: ('lh-,J! (.. .. ' 1,,c_ld...._;;; ;;i,-~;..,: f\. . ,<, ' Jn-p/j J/t,..-,,! a..., .... , .i, -r, !4,._, + "'':;-,. r.- t:iO Bt)Wtl !.tn tt~ ,uinulue ft- ::- I 'C I ~ ~. tte llertt0e outs,i..d~ t h~ ~!.~l$ /-el=- I'- null OQI~rplo;ii UiU?atbe.n, aueh th$ct ( I - I~ I ) { I- I-', I ) > o , l.fit A 4..il}~: ~4 1 , bo non• po1~,,om1tll fi' (r:) = fl,<~ ;1{ C-+ ~:, -2: f -1--J.-.?(-J:-J{ r-t-~JI«', ~.}::;-~ wheft /Y'1 ts Mj~ r-*si tiv-e !l1ttl@e1" " hA.e. no ze!'Os 1t1 the .armulu (fj ~ Ji:. I~ R .· r,;e note t h~rt thooratn 11 1.s v &11d or we e:hmig~ t k:.e hypothe.608 ' on ,<1 d,, r * /ti</>'-' I«, /:>J.,/,-/4:./ to retid 6/ / c Jc 2- --- .;;e get new 1-esul t,s. It 'lil." 8 t ttd!e liere {y 1 ~;'f,::: m I z: Thts g1vos ~ the 'lheO.r$n 12.. Under t he cypotl'leties (Ori tl:e ~., -< u J. ) ~t' t.,~~1 11 11 rt nv positive 1nt&~~r. t 11en t.l1e pol7ln@m1.a l I ~ ~~ f{ {e: J .=- /'loLcJ(~·, i!-) ~ >/' ( -<-) (,,,.,,, ! Z:-} ;,,_ 5 o4 r .1\sS / r-1.::::. P. except ~ =-o .• no g,e?OE within or on tl:ie e1rele To e ri,n: '.f th:1s vto h'9N17 1ailie the .-L to be equal to I!!- , et_ll L."'l the t..lteo.J'Gri! ot' Ob~khof':f arb1traril-y lmlt1ll nw.nb•~• fi<?tor out the ,,.,._ I e- -,tel&ng us that (/{f z-):: ~ no ,,, 1tte1>o-s w1 t tsi n or on If we t"Jrl,te c-=-0 P(,1 z-J t- >-Pfo1.-e-) a1; f' {-iJ-J = / ~ ,~ P.. Z J, · r ~~=~ 1e ~ zero of ~ /2:-) ~ a, f- > J," or 1f and m11,- if r,et u t-; oxcept ;os~1bl7 c.' ·/ ( r -=:... o , b. 4- •J · , wa have u.~t / a- I-= fl.- e~eept wk,_ no t6$rotl w1 t hin or on 1~he elre1e ,fi 4t'1-,.. rl ""' 12- o<t>t::!." "rlien -~ ~ ts a z~l'O of_ r;R /a J. • • ' -~ %,.._,Jl:i,Vfi. -~ t '•' \ ·'-· .. •• t:f7(2:-J tr:& all !ts ~JoOI"OB. out511.e t he circle t.~1 tipl-y"b',i,g bf (/(/2--J ~ d; ~ ~ , z {o1;~ ~ ol;, -+ ,·=a 1~ J , I ,,,.,, J .-.-< ,..)6 - ~ ' ol".. =R , under the a.ddi t1ona1 .;· h,rooth$1#. Thia new 101,nor11al itself hfw all lts zei"OS outside 1.em.d hence t he eo-.1--ell 111--y ~ '1 ·b () a.pr;,lied to 1t. It' iiro proe&ed ; , s.;n. tl;is ¼t&y 'fte otitnir1 tinclly t h& f!.~2~1 lri• de~· fl" Lot . be s ~ PI~ I ~ LJ ~c· 2- c. ,.. ::..J ,, w!:ch &ll 1 tfl a:J r.os o-utside t he circle f'ylrig t h.e- l'.'~)1$ t:10!":.S ., { I- IP!J ){ I- //J..,._J) > • p~ -1- :r.,,,., I~,_/ :a. <ft.,.,. t.lw pol7tHltttitd!. tf'... tz) = ; ; ~•.::.d ./L .I e1~, _,_ A.' :.1 h&ve all t heir z,e ros outside t he ei:rele I.f we wke / ; , ,.,,,., -}- ,,.A /I'-- ~ -t,... J ?H ~ ~ I p,... I.,_ ef 41 .,t:- D 11, (1..A- ~ o r =-'"' z..,, ••• J ~ o rr rfl: / .J -::.I{ • I>-.,.. I '> I ,, ,,(,,,_ ¥- 0 - ,, m.1d let ,;;... b:& on:v l~ i t1ve integer t I AU1 polj11.0tt:1al o~ M"Cl'GOVer re·ad / 8ft. / =- f/J-: I S(;) tl;tlt t!::~ ~~ !$ n.ow c itn be :mt:tt1,e ± J,c~~ ,· 'rl~:rem ••• -14. • ~·::::.o wr!th 111)" !ta ze~ outst~ the ei~le m.tt,-. tl'.~n tl:le ~l,nom!alc .,._ .S' Ifs t2-) ~ L ,-=-<,J to ff { ~/1-, . -f- >-~71 JI,< 2:-' )l ~ I 'h~ all tbtJ1lf S'ta~ B outsi&1 t."n.e c1:rel.e (?:-/ ~I?. ... /:z,./t::.~, Theorem 9 can be written to reach t:4· l ,· f'(-z-,1 ~ !,et ,·-=J be a pol-y:nomial or degree """' wtth ~ all its zeros outside /~ /-== K Let r, be arbitrary pc4d.t1v• Qn Then the polwn,om1a1~ ,0 ( Q(tJ ::. .J..< hae al\ it$ ze:roe outside / .2:- l =- I( The pl'f>Ot' given above shows that P t .1. J I ~ 1-=.R • ~ ~ M...;.. { /~, IJ ... J I""" I) where Let are the zeros of such that has no zero• within /?f.a} • Then tJt' -j' ~ tG- 7 o and we erm t1nd a zeros e:ll outside {i?I-:... 17 then I' I .a J has 1 ts haa no ze:roe \V1th1n /J/•.;1 f' , ar1d hence emmot have ony zero, on If,} ft'} r K < (-' < -<. Then by the proof of theorem 9 given abo·we since /a/-::. t,1 1 ., • •• ~ ~........ /:etc.~ • Hence must have all its zeros outeide /r/ ~ ~, Theorem 10 ean be :rewr1 ,. tten to reach Let Pf~I~ 2 J,.- ~ i be- a pol-ynom:tal of Otpeo l'k with ~--=o all its zeros outslde /?;/~ I:. • be a Let sequence of :arbitrary po.e1t1VEI numbers n11>111:l.als Ifs ( 'c J ::; z ,'=-1> f f (,. i -,,t<,. - ,,_ ) Jl-::.../ have all their zeros outs14e / ':/r/-:::. I< The proof 1a a.'l"l. easy induction on J.- z L ~ .S ltto· ~ d ~ ~ • ·:-.a ~k~&ftiic~ ~ ! ! ~ , -At tftkxfcl~ t ~ . - •Sa ~ #G~ ~ l 1 ' b f , ~ ~ ~ tium ~ k D)3 SUED..(lr'.V .of ·•a · :n._· M .---. nnt ~~ na~ta ~ld,A' • ~ (J'6 ~ 1 a- j -J '!110 p1"0bl,~ 1 ot t he d(tte:::~li1,at!en O,f the beet V~1U<t of R ,t o~ a given pol-ynoo'i:1 al wtth ti non- c'Oro e.on.at~n.t term 1e next t cQk&n At1 upper 11ml t tor poa~:U.:il . fJ VtilUOO of" up., R 1s ~ - - tm4 t1.eee:aB~l"J !fn.d s t.:d: f '1c1er:i t eondt t1<tt'1S er-e ,g iven .for t Jt&t upp$',_. 1 lwi t te b" &.J;:pros el'lOd arbi. t~lJPily .elt:'H~el1r. wh1eb estht:&te R ft~$ Vari ous f'G~rula-e: e!tod. f ~ t b,:!J literature.,. ~,er limlte :.ror t he 1.t0du11s of s 01'()s of polyncrn-:;i G.l s are e1 t$d t-lnd t he1i- /(tt!Qy c;tl1e.?t b 1:,0 ,. ~, 1-~ ,._ I J res,ul.~, 1t 1a s l:,orn tbnt 1f' 1iil,.._l-I 1t& ~ et"Otll Oi.1t61d-0 I el:::. If • 'I; I?, /'f•I == Z- J,. ~ ,· ,.•:: c) • t Jion ao :ttss -t') ' l z. <. ,~::.; :If' ,,At ~ . · · ~ fa s ,, • • • is ll se-q'UG"JH.'!G b ere t hen tJ'le po.l}nW.ids t; ff ( :ii-A,.->t }/,«· r ,· (s = ,, ~.., ... J P..loo 1r ol. 1,, _... J ,IA/ • • &.mi 1a &rls,' eequen ee of complex 11w1it ers r, ~ . . . , -;,,_,,. .. ar-a im-y oor-apl -e:x nm;1bers ~1 t us ted on t he uni t, e1.r el a t hen t;Le polyno ,.,i r::le . !rs ri!: I ;:_ ~ TT ( o(/1 t > I~; I?..),&. ~ t" L" . i'= ,, .ll ::: , ,IL. ~ h,s.ve t'.1ll t he1r :!e-ros out~! c,~ t .t o eirelo / 'i:t .!:: K.